Finite Group Databasebeta

← #10694701

Subgroups — #10694701

Order 768.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5116C211yesmaximal, characteristic
2C3×#1808C211yesmaximal, characteristic
3C8×#217C211yesmaximal, characteristic
4#69529C212yesmaximal
5#69510C212yesmaximal
6#9782—31nomaximal, Sylow
7C23×C3×C8C2211yescharacteristic
8C2×#924C2212yes—
9C22×Dic12C411yescharacteristic, complemented
10C2×C3×#107—21no—
11C3×#380C2212yes—
12C4×#217C2211yescharacteristic
13C2×#830C2211yescharacteristic
14#5116—21no—
15#5116—21no—
16#5116—21no—
17#5116—21no—
18C2×#818C411yescharacteristic, complemented
19C2×#816C412yescomplemented
20C2×C8×Dic3—21no—
21C3×#340—21no—
22#4550—21no—
23#4527—24no—
24#4452—24no—
25C2×#467—31no—
26#1996—32no—
27#1808D311yescharacteristic
28C8×C22⋊C4—31no—
29#1520—32no—
30C23×C3×C4C2311yescharacteristic
31C22×Dic6C2×C412yes—
32C22×C3×C8—42no—
33C22×C3×C8C2×C411yescharacteristic
34C22×C3×C8C2×C411yescharacteristic
35C22×C3×C8D411yescharacteristic
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C2×C3×C4⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×#217—22no—
42C2×#217D412yes—
43C2×C4⋊Dic3C2×C412yes—
44#924—22no—
45#924—22no—
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—22no—
49C2×C4×Dic3—21no—
50C22×C3⋊C8—21no—
51C2×Dic12—21no—
52C2×Dic12—21no—
53C2×Dic12—42no—
54C3×#107—42no—
55C3×#95—22no—
56C3⋊#95—21no—
57#830—21no—
58#830D412yes—
59#818—21no—
60#818—21no—
61#816—22no—
62#816—22no—
63C8×Dic3—42no—
64C22×Q16—31no—
65C23×C8C2×D311yescharacteristic
66C2×C22⋊Q8—32no—
67#467—61no—
68#467—61no—
69#467—61no—
70#467—61no—
71C2×#107—21no—
72C2×#107—31no—
73C2×#103—32no—
74C2×C22⋊C8—31no—
75C2×C4×C8—61no—
76#380C2×D312yes—
77C4×C22⋊C4—31no—
78#342—64no—
79#340—21no—
80#336—61no—
81#328—64no—
82C24×C3C2×D411yescharacteristic
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—42no—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4111yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—22no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—42no—
98C2×Dic6—42no—
99C2×C3×C8—22no—
100C2×C3×C8—22no—
101C2×C3×C8—22no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8C22⋊C411yescharacteristic
106C2×C3×C8C22⋊C411yescharacteristic
107C2×C3×C8C22⋊C411yescharacteristic
108C2×C3×C8C22⋊C411yescharacteristic
109C3×C4⋊C4—42no—
110C3×C4⋊C4—42no—
111C3×C22⋊C4—44no—
112#217—22no—
113#217—42no—
114#217—42no—
115#217Q1614yescomplemented
116C4⋊Dic3—22no—
117C4⋊Dic3—22no—
118Dic3⋊C4—42no—
119Dic3⋊C4—42no—
120C4×Dic3—41no—
121C4×Dic3—41no—
122C2×C3⋊C8—22no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8—41no—
125Dic12—42no—
126Dic12—42no—
127C22×Q8—32no—
128C23×C4C22×D311yescharacteristic
129C2×Q16—61no—
130C2×Q16—61no—
131C2×Q16—122no—
132C22×C8—42no—
133C22×C8—61no—
134C22×C8C4×D311yescharacteristic
135C22×C8C4×D311yescharacteristic
136C22×C8D1211yescharacteristic
137C22×C8D1211yescharacteristic
138C22×C8#6711yescharacteristic
139C22×C8#6711yescharacteristic
140C22⋊Q8—62no—
141C22⋊Q8—62no—
142C22⋊Q8—62no—
143C22⋊Q8—62no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—32no—
146C2×C4⋊C4—62no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—32no—
149C2×C22⋊C4—62no—
150C2×C42—61no—
151#107—42no—
152#107—61no—
153#107—61no—
154#103—62no—
155#103—62no—
156C22⋊C8—32no—
157C22⋊C8—61no—
158C4×C8—122no—
159#95—22no—
160#95—61no—
161C23×C3—41no—
162C23×C3—41no—
163C23×C3C4×D411yescharacteristic
164C23×C3#12111yescharacteristic
165C23×C3#12711yescharacteristic
166C23×C3C2×Q1612yes—
167C23×C3#13512yes—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—82no—
178C2×C3×C4—82no—
179C2×C3×C4—82no—
180C2×C3×C4C2×C22⋊C411yescharacteristic
181C2×C3×C4C4×D411yescharacteristic
182C2×C3×C4#12111yescharacteristic
183C2×C3×C4#12411yescharacteristic
184C2×Dic3—24no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191Dic6—42no—
192Dic6—42no—
193Dic6—82no—
194C3×C8—21no—
195C3×C8—21no—
196C3×C8—21no—
197C3×C8#9912yes—
198C3⋊C8—42no—
199C24D3×D411yescharacteristic
200C2×Q8—62no—
201C2×Q8—62no—
202C2×Q8—122no—
203C2×Q8—122no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4C2×C4×D311yescharacteristic
211C22×C4C2×D1211yescharacteristic
212C22×C4D3×D411yescharacteristic
213C22×C4#23911yescharacteristic
214C22×C4#23911yescharacteristic
215C22×C4C2×#6711yescharacteristic
216Q16—122no—
217Q16—122no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—62no—
225C2×C8—121no—
226C2×C8—121no—
227C2×C8(C2×D3)⋊C411yescharacteristic
228C2×C8(C2×D3)⋊C411yescharacteristic
229C2×C8(C2×D3)⋊C411yescharacteristic
230C2×C8(C2×D3)⋊C411yescharacteristic
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—122no—
236C4⋊C4—122no—
237C22⋊C4—34no—
238C22⋊C4—44no—
239C22⋊C4—62no—
240C22⋊C4—122no—
241C22⋊C4—122no—
242C42—121no—
243C42—121no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3#39011yescharacteristic
253C22×C3C4×Q1612yes—
254C22×C3#46712yes—
255C22×C3#49411yescharacteristic
256C22×C3#49511yescharacteristic
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—162no—
261C3×C4C2×#9911yescharacteristic
262C3×C4#41011yescharacteristic
263Dic3—44no—
264Dic3—82no—
265Dic3—82no—
266C23—41no—
267C23—41no—
268C23#89311yescharacteristic
269C23#89511yescharacteristic
270C23#91212yes—
271C23D3×Q1612yes—
272C23#94011yescharacteristic
273Q8—122no—
274Q8—122no—
275Q8—242no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—64no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—242no—
294C2×C4—242no—
295C2×C4#89311yescharacteristic
296C2×C4#89511yescharacteristic
297C2×C4C2×(C2×D3)⋊C411yescharacteristic
298C2×C4#94711yescharacteristic
299C8—21no—
300C8—21no—
301C8—21no—
302C8—122no—
303C8#80612yes—
304C2×C3—22no—
305C2×C3—21no—
306C2×C3—22no—
307C2×C3—21no—
308C2×C3#196411yescharacteristic
309C2×C3#196911yescharacteristic
310C2×C3#197611yescharacteristic
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—42no—
318C22—42no—
319C22#487312yes—
320C22#488712yes—
321C22#498811yescharacteristic
322C22#515511yescharacteristic
323C22#518311yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—124no—
328C4—162no—
329C4—242no—
330C4—242no—
331C4C2×#80611yescharacteristic
332C4#500611yescharacteristic
333C3#978211yescharacteristic, complemented, Sylow
334C2—22no—
335C2—21no—
336C2—22no—
337C2—21no—
338C2#6952811yescharacteristic
339C2#7062711yescharacteristic
340C2#7065811yescharacteristic
subgroupG/H
centerC22#4988
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1808C2
socleC22×C3#390
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1808D3
hypocenterC3#9782
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1808

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
9. C22×Dic12C4semidirect
18. C2×#818C4semidirect
19. C2×#816C4semidirect
115. #217Q16semidirect
333. C3#9782semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#51162
C3×#18082
C8×#2172
#695292
#695102
#97823